Ceramic matrix composite (CMC) materials are used for high-temperature components such as gas turbine blades, vanes, and shroud surfaces. The walls of these components have a front surface that optionally may be coated with a ceramic insulating material and that is heated by the turbine combustion gas, and a back surface that is cooled by a cooling air flow. Cooling is accomplished by any of several conventional methods. For lower temperature applications, laminar backside cooling is effective; however, entry points for cooling air flow tend to have locally high heat transfer coefficients. For higher heat flux conditions, more aggressive cooling methods are required, including, for example, impingement cooling. Typically, impingement cooling is accomplished by directing jets of the cooling air toward the back side of the CMC wall in order to remove heat energy and to lower the temperature of the CMC material. For high thermal conductivity CMCs such as melt-infiltrated SiC/SiC composites and others, the adverse side effects from such impingement cooling are negligible. However, for low thermal conductivity CMCs such as the oxide-oxide classes of materials, the impingement method results in high in-plane thermal gradients on the cooled surface. Improved techniques for cooling ceramic materials used in high temperature applications are thus desired.